Using analytical techniques, it has been proved in [1]}
that, for spherically symmetric hairy black-hole spacetimes whose
external matter fields are characterized by a non-positive
energy-momentum trace [2]}, the radius \(r_{\gamma }\)  of the innermost null circular
geodesic of the curved black-hole spacetime is bounded from above by the remarkably compact relation [3]}
\(r_{\gamma }\le 3M\ ,\) 
